Question

The overall reaction in the lead storage battery is \(\mathrm{Pb}(s)+\mathrm{PbO}_{2}(s)+2 \mathrm{H}^{+}(a q)+2 \mathrm{HSO}_{4}^{-}(a q) \longrightarrow\) \(2 \mathrm{PbSO}_{4}(s)+2 \mathrm{H}_{2} \mathrm{O}(l)\) Calculate \(\mathscr{E}\) at \(25^{\circ} \mathrm{C}\) for this battery when \(\left[\mathrm{H}_{2} \mathrm{SO}_{4}\right]=4.5 \mathrm{M}\), that is, \(\left[\mathrm{H}^{+}\right]=\left[\mathrm{HSO}_{4}^{-}\right]=4.5 \mathrm{M} . \mathrm{At} 25^{\circ} \mathrm{C}, \mathscr{E}^{\circ}=2.04 \mathrm{V}\) for the lead storage battery.

Short answer

Using the Nernst equation and the given reaction conditions, we can find the cell potential (EMF) of the lead-acid battery at 25°C. The number of electrons transferred in the reaction is 2, and the reaction quotient Q is \(\frac{1}{[\mathrm{H}^+]^2 [\mathrm{HSO}_4^-]^2}\) which simplifies to \(\frac{1}{(4.5)^2(4.5)^2}\). Plugging all values into the Nernst equation, we find that the cell potential (E) is approximately 2.01 V.

Step-by-step solution

Write the Nernst equation

To find the EMF of the cell, we must use the Nernst equation, which relates the cell potential, standard cell potential, temperature, the number of electrons transferred, and the concentrations of reactants and products. The Nernst equation is: \[E = E^\circ - \frac{RT}{nF} \ln{Q}\] Where: - E is the cell potential (EMF) - E° is the standard cell potential (2.04 V, given) - R is the gas constant (8.314 J/mol·K) - T is the temperature in Kelvin (25°C = 298 K) - n is the number of electrons transferred in the reaction - F is the Faraday constant (96485 C/mol) - Q is the reaction quotient for the reaction at the specific concentrations

Find the number of electrons transferred n

Analyzing the given reaction, determine the number of electrons transferred by balancing the charges of reactants and products: \(\mathrm{Pb}(s)+\mathrm{PbO}_{2}(s)+2 \mathrm{H}^{+}(a q)+2 \mathrm{HSO}_{4}^{-}(a q) \longrightarrow 2 \mathrm{PbSO}_{4}(s)+2 \mathrm{H}_{2} \mathrm{O}(l)\) 2 electrons are transferred in the overall reaction. Thus, n = 2.

Calculate the reaction quotient Q

For the given reaction, the reaction quotient Q can be defined as: \[Q = \frac{[\mathrm{PbSO}_4]^2 [H_2O]^2}{[\mathrm{H}^+]^2 [\mathrm{HSO}_4^-]^2}\] Since PbSO₄ and H₂O are both solids, their concentration terms are omitted from the Q calculation: \[Q = \frac{1}{[\mathrm{H}^+]^2 [\mathrm{HSO}_4^-]^2}\] We are given that [H⁺] = [HSO₄⁻] = 4.5 M. Plug in these values to calculate Q: \[Q = \frac{1}{(4.5)^2(4.5)^2}\]

Calculate the cell potential E

Now plug all the values into the Nernst equation: \[E = 2.04 - \frac{8.314 \times 298}{2 \times 96485} \ln{\frac{1}{(4.5)^2(4.5)^2}}\] Calculate the value of E: \[E \approx 2.01 \, V\]

Write the final answer

The electromotive force (EMF) of the lead-acid battery at 25°C and given concentrations is approximately 2.01 V.

Key concepts

Electrochemistry

Electrochemistry is a fascinating branch of chemistry that studies the interactions between electrical energy and chemical changes. It is crucial for understanding how batteries function, including how electricity can be generated by chemical reactions or how electrical energy can drive chemical changes.
This discipline focuses on redox reactions (reduction-oxidation), which involve the transfer of electrons between substances.

• In oxidation, a substance loses electrons, increasing its oxidation state.
• In reduction, a substance gains electrons, lowering its oxidation state.

Electrochemical cells, like galvanic cells, utilize these redox reactions to generate electrical energy. By harnessing the movement of electrons from one substance to another, we can create a flow of electric current, which is the basis for all battery-powered devices.

Lead Storage Battery

The lead storage battery, also known as a lead-acid battery, is one of the oldest and most important types of rechargeable batteries. Its wide range of applications includes use in cars, emergency lighting, and storage for renewable energy.
Within the battery, there are two main electrodes: a lead electrode and a lead dioxide electrode, immersed in a sulfuric acid solution. As the battery discharges, lead and lead dioxide react with sulfuric acid, forming lead sulfate and water.
Important features of lead storage batteries:

• Cost-effective and reliable for large-scale energy storage.
• Maintenance is essential due to their potential to lose capacity or degrade over time.
• Heavy and contain toxic lead, requiring careful handling and disposal.

To recharge, the chemical reactions are reversed by applying an external electrical current, restoring the lead and lead dioxide electrodes.

Cell Potential Calculation

Calculating cell potential is crucial for optimizing the performance of electrochemical cells. The Nernst equation is a powerful tool used to determine the cell potential under varying conditions, taking into account concentrations, temperatures, and more.
For a typical galvanic cell reaction:- The standard cell potential (\(E^0\)) represents the potential when all components are in their standard states.- The Nernst equation \(E = E^0 - \frac{RT}{nF} \ln{Q}\) requires the calculation of the reaction quotient \(Q\), considering products and reactants' concentrations.- \(n\) represents the number of electrons transferred in the reaction.
The Nernst equation connects electrochemistry with thermodynamics, showing how the potential depends on the reaction progression. Accurate \(E\) values can guide the design of better batteries and electrochemical devices, enhancing their efficiency and longevity.